Method for modelling the production of hydrocarbons by a subsurface deposit which are subject to depletion

ABSTRACT

A method for forming a model simulating production, by an underground reservoir subject to depletion, of hydrocarbons comprising notably relatively high-viscosity oils. From laboratory measurements of the respective volumes of oil and gas produced by rock samples from the reservoir subject to depletion, and the relative permeabilities (Kr) of rock samples to hydrocarbons, a model of the formation and flow of the gas fraction is used to determine a volume transfer coefficient (hv) by means of an empirical function representing the distribution of nuclei that can be activated at a pressure P (function N(P)) which is calibrated with reference to the previous measurements. Considering that the nuclei distribution N(P) in the reservoir rocks is the same as the distribution measured in the laboratory, the numerical transfer coefficient corresponding thereto in the reservoir at selected depletion rates is determined using the gas fraction formation and flow model, which allows predicting the relative permeabilities in the reservoir and the production thereof which is useful for reservoir engineering. Method for forming a model allowing to simulate the production, by an underground reservoir subjected to depletion, of hydrocarbons comprising notably relatively high-viscosity oils. From laboratory measurements of the respective volumes of oil and gas produced by rock samples from the reservoir and subjected to depletion, and the relative permeabilities (Kr) of rock samples to hydrocarbons, a model of the formation and flow of the gas fraction is used to determine a volume transfer coefficient (hv) by means of an empirical function representing the distribution of nuclei that can be activated at a pressure P (function N(P)) which is calibrated with reference to the previous measurements. Considering that the nuclei distribution N(P) in the reservoir rocks is the same as the distribution measured in the laboratory, the numerical transfer coefficient corresponding thereto in the reservoir at selected depletion rates is determined using the gas fraction formation and flow model, which allows to predict the relative permeabilities in the reservoir and the production thereof. Applications notably reservoir engineering.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for modelling the productionof hydrocarbons comprising notably relatively high-viscosity oils bypetroleum reservoirs subjected to decompression or depletion.

2. Description of the Prior Art

The development of hydrocarbon reservoir production simulation generallyinvolves several stages. Laboratory experiments are first interpreted.Then, the phenomena are modelled on the laboratory scale before anextrapolation is carried out on the reservoir scale. The quantitiesmeasurable on the laboratory scale and which have meaning on thereservoir scale therefore have to be determined (saturation, pressure,average concentration). The main requirement lies in the fact that themodel must describe, for the same rock-fluids system, with the sameparameters, experiments carried out under different conditions, that isfor different depletion rate changes, withdrawal rate changes, etc. Oneof the main parameters is the relative permeability (Kr) which expressesthe interactions between the reservoir fluids and the rock (FIG. 1). Inwater or gas drive methods, the relative permeabilities used forreservoir simulation are directly measured on cores (FIG. 2).

The mechanism of oil production from an underground hydrocarbonreservoir, by means of a decompression (well-known as solution gasdrive) has been used and studied for a long time in the petroleumsphere. This production mechanism, which essentially produces oilsaturated with light elements by depleting the reservoir, is eitherfavored as in the case of viscous oils or avoided in the case of lightoils, at least at reservoir production start, because it leads to anearly production of gas and to a low recovery rate. However, in anycase, modelling the reservoir production is necessary to control thismechanism.

Modelling of the production by depletion poses a specific problem fornumerical simulations. Unlike the water and oil drive productionmethods, the relative permeabilities Kr measured in the laboratory onsamples containing viscous oils cannot be directly used in numericalreservoir simulations. The reason is known and explained in manypublications: on the one hand, the diffusion mechanism of the lightconstituent contained in the oil phase to the gas phase (“offequilibrium” transfer) and, on the other hand, the gas flow in thediscontinuous form of bubbles or bubble strings. The consequence ofthese two effects is that the Kr values determined in the laboratorygreatly depend on the experimental conditions, among other things thedepletion rate (experiment duration). Another well-known method ofsimulating foam flows in a modelled porous medium, known as “PopulationBalance Modelling”, is described by Arora, P., Kovscek, A. R., 2001,Mechanistic Modeling of Solution Gas Drive in Viscous Oils, SPE 69717International Thermal Operations and Heavy Oil Symposium, Porlamar,Margarita Island, Venezuela, March 12-14. The method introduces a largenumber of parameters: nucleation rate, bubble coalescence rate, rate ofbubble formation during flow, which cannot be determined experimentally.

Pore network models are also known, which are notably described by Li,X., Yortsos, Y. C., 1991, Visualization and Numerical Studies of BubbleGrowth during Pressure Depletion, SPE 22589 66^(th) Annual TechnicalConference and Exhibition, Dallas, Tex., October 6-9, based on apore-scale physics and which therefore cannot simulate an experiment onthe scale of a core and take into account of the boundary conditionsspecific to the experiments. These models have been tested only forlight oils and they do not take into account dispersed gas flow.

The model described by Tsimpanogiannis, I. N., Yortsos, Y. C., 2001, AnEffective Continuum Model for the Liquid-to-Gas Phase Change in a PorousMedium Driven by Solute Diffusion: I. Constant Pressure Decline Rates,SPE 71502 Annual Technical Conference and Exhibition, New Orleans, La.,30 September-3 October, is a model using continuous equations. It allowsgood understanding of the mechanisms involved in depletion production(solution gas drive): number of nucleated bubbles, maximumoversaturation, and their influence on the critical gas saturation. Onthe other hand, it uses a large number of parameters that cannot bedirectly measured, such as the number and the size of the bubbles.Furthermore, this model does not deal with the flow of the phases andthe mass transfer throughout an experiment.

The model described by Sheng, J. J., Foamy Oil Flow in Porous Media, PhDDissertation, University of Alberta, Edmonton, Canada, takes intoaccount the equilibrium delay due to the growth and to the transferbetween a dispersed gas and a continuous gas by means of exponentiallaws as in a chemical reaction. This method is also used in anindustrial simulator (STARS). Such a solution does not show the physicsof the phenomenon. It is difficult to interpret experiments in terms ofphysical parameters and therefore to be predictive. This approach takesinto account a dispersed gas phase and a second, continuous phase. Againin this case, transfer between the two phases is governed by a chemicalreaction type equation. Calibration is performed by adjusting parametersof the chemical reactions, parameters which are based on no physicaljustification. It is therefore impossible to predict parameters underreservoir conditions.

In general terms, no known model takes into account, within the scope ofthe solution gas drive process, and in a continuous approach, all of themechanisms by allowing calculations under the reservoir flow conditionsby using laboratory experiments.

The method according to the invention allows, from laboratorymeasurements on such samples and by means of suitable correctionsdescribed hereafter, realistic modelling of the production of a depletedreservoir, whatever the viscosity of the oils produced, and moreparticularly when it contains viscous oils, by using a compositionalreservoir simulator available on the market.

The modelling method according to the invention allows simulation ofproduction by an underground reservoir under the effect of depletion. Itaffords an excellent compromise between the accuracy to the physicalmechanisms and modelling simplicity, in particular a small number ofparameters that can be determined from a single laboratory experiment.

The method essentially comprises the following stages

-   a) measuring in the laboratory respective volumes of oil and gas    produced by rock samples from a reservoir and subjected to    depletion, as well as relative permeabilities of rock samples to    hydrocarbons,-   b) determining, by a gas fraction formation and a flow model, a    volume transfer coefficient by means of an empirical function    representing the distribution of microbubbles or nuclei as a    function of the pressure that is calibrated with reference to the    previous measurements, and-   c) while considering that the distribution of microbubbles or nuclei    in the reservoir rocks is the same as the distribution of the    microbubbles deduced from the laboratory measurements, determining,    by means of this gas fraction flow model, the numerical transfer    coefficient that corresponds thereto in the reservoir at selected    depletion rates, which allows prediction of the relative    permeabilities in the reservoir and the reservoir production.

According to a preferred embodiment, the gas fraction flow model isessentially described by a parameter F characterizing the force requiredfor untrapping the bubbles; a parameter α characterizing the change ofthe gas phase to the continuous form, the two parameters beingdetermined by calibration from the laboratory measurements, and by thevalues of the relative permeability to the continuous gas fraction.

In the model obtained with the present method, the transfer is modelledby a volume transfer coefficient which has meaning on the laboratoryscale and on the reservoir scale, whose dependence has been expressed asa function of the various parameters: gas saturation, oversaturation,liquid velocity.

By means of a two-stage procedure structured on a common significantparameter characterizing the nucleation of the gas phase, which is validfor the experimentally studied samples as well as for the rocks of thereservoir, the first stage being carried out with reference tolaboratory measurements, it is possible to construct a predictivemodelling tool allowing realistic representation of the conditions offlow of the viscous fractions of the oil in place in the reservoir.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the method according to the inventionwill be clear from reading the description hereafter of a non limitativeembodiment example, with reference to the accompanying drawings wherein:

FIG. 1 illustrates the principle of a petroleum reservoir productionsimulation, the main useful parameter being the relative permeabilitywhich expresses the interactions between the fluids (water, oil or gas)and the rock,

FIG. 2 shows, for water or gas drive methods, the experimental schemeallowing to obtain, from measurements on samples, relativepermeabilities Kr suitable at the laboratory stage as well as in thereservoirs,

FIG. 3 illustrates the principle of determination of the characteristicparameters of flow of an oil by depletion from laboratory experiments,which is the object of the first essential stage of the method,

FIG. 4 shows the principle of use of a flow simulator for carrying out anumerical experiment under reservoir conditions allowing to determine“reservoir Kr” values, which is the object of the second essential stageof the method,

FIG. 5 diagrammatically shows the various “pseudo”-stages present in theporous medium (the residual water phase is not mentioned but it exists),

FIG. 6 shows simulation examples for a light C₁-C₃-C₁₀ oil,

FIGS. 7 and 8 show a first series of simulations carried out fordifferent viscous oils (250 cp and 3300 cp) in the same rock type, and

FIGS. 9 and 10 show a second series of simulations, the first one withan oil whose viscosity is about 1500 cp at 0.5 and 12 barj⁻¹, the secondwith an oil whose viscosity is about 300 cp at 0.8 and 8 barj⁻¹.

DETAILED DESCRIPTION OF THE INVENTION

A first important point of the method of the invention relates to the“off-equilibrium” aspect of the light component transfer. It is based onmodelling of the gas phase nucleation allowing prediction of the densityof the bubbles and the pressure at which they appear. A law ofdistribution of the number of pre-existing “nuclei” or microbubbles as afunction of the pressure is suggested. This empirical law N(P) takesinto account the properties of the solid (surface roughness), theproperties of the fluids and the physico-chemical interactions betweenthe fluids and the solid (wettability for example). A relation form, forexample exponential or power law, is imposed from the publishedmeasurements and the few parameters of this law (threshold pressure,exponent of the power law) are determined from the experiment bycalibration. This law is considered valid at the laboratory stage aswell as at the reservoir stage. From knowledge of this law N(P) and ofthe thermodynamic properties of the fluids (known properties), themethod comprises a computing stage allowing determination of thetransfer between the phase of the light component between the liquid andthe gas. This computation takes into account the off-equilibriumdifference and it therefore allows prediction of the evolution of thegas production with time, for any depletion rate.

-   -   The second point of the modelling method relates to the flow of        the gas in a non-continuous form. Three possible situations for        the gas are distinguished: either a phase trapped in form of        bubbles or “bubble strings”, or a mobile dispersed phase carried        along by the oil flow, or a continuous phase flowing according        to the conventional laws relative to flows in porous media        (Darcy's law).

Based on known results in untrapping and bubble flow physics, the methodallows producing a gas flow model described by a very small number ofparameters that can be either calibrated on depletion experiments ormeasured separately:

-   -   a parameter F characterizing the force required for bubble        untrapping (adhesion to the walls or capillary trapping), to be        determined by calibration,    -   a parameter α characterizing the change of the gas phase to the        continuous form. It has been shown by several authors that the        saturation at which the gas goes into the continuous form Sgc is        a law expressed as a power of the depletion rate. Parameter α is        the exponent of this power law, assumed to be the same for a        sample and a given oil, whatever the experiment conditions, to        be determined by calibration also, and    -   the values of the relative permeability to the continuous gas,        measured by conventional injection drive methods.

The flow model provided allows calculation of the flow properties(critical saturations, gas flow, etc.) as a function of constants F andα, of the properties of the fluids and of the experimental conditions(velocity of flow, depletion rate, etc.).

Coupling of the transfer model with the flow model allows simulation ofan experiment in any condition. It is used in two stages respectivelyillustrated by FIGS. 3 and 4:

-   1) with the conditions of the experiments carried out in the    laboratory, determination of the characteristic parameters F, α and    N(P) by calibration (modification of the parameters until an    agreement is obtained between the real and the simulated    experiment),-   2) with the reservoir conditions, predictive operation that is    “numerical” experiment that can be carried out at very slow    depletion rates for example. The “reservoir” relative permeabilities    are then determined by means of a standard calibration method,    exactly as for a real experiment.

Transfer Function Dependence Determination

Growth by diffusion in the case of a depleted liquid is controlled bythe concentration gradient at the surface of the bubble. In a continuousapproach, this local gradient is not accessible and it is replaced by asurface transfer coefficient h_(s). The transfer flow density is assumedto be proportional to the difference between the equilibrium valueC_(eq) at the bubble interface and the average concentration C in theliquid. Transfer coefficient h_(s) allows to calculate the flow densityφ:φ=h _(s)(C−C _(eq))  (1)with φ (mol.m⁻².s⁻¹), h_(s) (m.s⁻¹). Introduction of a transfercoefficient to replace a local gradient is a relatively common procedurein physics.

Hereafter an expression for h_(s) as a function of characteristicquantities in the case of the growth of a spherical bubble population inan infinite medium is determined.

A fluid volume V (liquid+gas) is considered. The pressure in the gas isP. The total surface area of the bubbles in this volume is denoted by sand N₀ is the total number of bubbles per volume unit of fluid. All thebubbles are assumed to have the same radius r.

Total volume of the bubbles: $\begin{matrix}{V_{G} = {N_{0}V\frac{4\pi\quad r^{3}}{3}}} & (2)\end{matrix}$

Surface area of the bubbles:s=N ₀ V4πr ²  (3)

The radius can be eliminated by expressing the surface area as afunction of the volume: $\begin{matrix}{s = {N_{0}{{V4\pi}\left( \frac{3V_{G}}{N_{0}{V4\pi}} \right)}^{2/3}}} & (4)\end{matrix}$

By definition of the flow surface density: $\begin{matrix}{\frac{\mathbb{d}n}{\mathbb{d}t} = {\varphi \cdot s}} & (5)\end{matrix}$

Henry's law:C _(eq) =k _(s) P  (6)

An equation for spherical bubbles is then obtained as follows:$\begin{matrix}{\frac{\mathbb{d}n}{\mathbb{d}t} = {h_{s}N_{0}{{V4\pi}\left( \frac{3V_{G}}{N_{0}{V4\pi}} \right)}^{2/3}\left( {C - {k_{s}P}} \right)}} & (7)\end{matrix}$

An estimation of the surface transfer coefficient h_(s) can be given byreplacing the gradient at the wall in the local approach by a meangradient, using the mean distance d between bubbles $\begin{matrix}{h_{s} \approx \frac{D}{d}} & (8)\end{matrix}$

The mean distance between bubbles is expressed as a function of thenumber of bubbles N₀ per unit volume:d ³=1/N ₀  (9).

Hence finally: $\begin{matrix}{\frac{\mathbb{d}n}{\mathbb{d}t} = {{DN}_{0}^{1/3}N_{0}{V4\pi}\left( \frac{3V_{G}}{N_{0}{V4\pi}} \right)^{2/3}\left( {C - {k_{s}P}} \right)}} & (10)\end{matrix}$

-   -   and, if simplified: $\begin{matrix}        {\frac{\mathbb{d}n}{\mathbb{d}t} = {{aDN}_{0}^{2/3}V^{1/3}{V_{G}^{2/3}\left( {C - {k_{s}P}} \right)}}} & (11)        \end{matrix}$    -   where a is a constant        a=(4π) ^(1/3)3^(2/3)≈4.84  (12)

Changing to Darcy's Scale

On Darcy's scale, the inner surface of the bubbles is not known.Therefore a “volume” transfer coefficient h_(v) defined as a function ofthe flow of moles per volume unit of fluid is defined as:Φ=h _(v)(C−C _(eq))  (13).

The dimension of h_(v) is (time)⁻¹. In order to show the dependence ofh_(v) as a function of the various “microscopic” parameters of theexperiment, this law is identified with the result of the previouscalculation, Equation (11):Φ=1/V dn/dt=h _(v)(C−C _(eq))  (14)

-   -   hence: $\begin{matrix}        {h_{v} \approx {{aDN}_{0}^{2/3}V^{{- 2}/3}V_{G}^{2/3}}} & (15)        \end{matrix}$

The gas saturation (S=V_(g)/V_(total)) can also be introduced:h _(v) ≈aDN ₀ ^(2/3) S _(G) ^(2/3)  (16)

It has to be noted that this result is obtained with a greatlysimplified model of equidistant bubbles of uniform size. But it allowsexplaining the dependence as a function of the various parameters: gassaturation, bubble density and molecular diffusion. In practice, theprefactor as well as the powers can be adjusted.

We thus have a relation that gives the evolution of the number of gasmoles. In problems related to porous media, it is more demanding to workwith variables such as saturations. Using the perfect gas law allowsshowing the gas saturation rather than the number of mole. The perfectgas law gives: $\begin{matrix}{n = \frac{{PV}_{g}}{RT}} & (17)\end{matrix}$

Therefore substituting n in Equation (14) provides: $\begin{matrix}{\frac{\mathbb{d}\left( {PS}_{G} \right)}{\mathbb{d}t} = {h_{v}{{RT}\left( {C - C_{eq}} \right)}}} & (18)\end{matrix}$

A continuous equation is obtained which gives the evolution of the masstransfer between a fluid saturated with light elements and the gasphase. It involves, which is an important point of the approachselected, only mean variables which have a physical meaning in Darcy'sapproach.

It is seen that the volume transfer coefficient h_(v) first depends onthe number of bubbles, which itself depends on the oversaturation. Inorder to determine from the experiments this transfer coefficient bymeans of the calibration technique the results obtained on the finerscale of Relation (7) are used.

Nucleation is an important mechanism and, on this scale, the only meansto take it into account is to introduce a site size distribution. Inthis model, this amounts to making N₀ dependent on oversaturation ΔP. Inthe model, the approach described by Yang, S. R., et al., 1988, Amathematical Model of the Pool Boiling Nucleation Site Density in termsof the Surface Characteristics, International Journal of Heat and MassTransfer, 31(6), 1127-1135, is used by introducing an exponential law:$\begin{matrix}{N_{0} \propto {\exp\left( {- \frac{\delta}{P - P_{eq}}} \right)}} & (19)\end{matrix}$

However, this equation has to be modified in order to take into accountof the oversaturation threshold ΔP_(threshold):

-   -   N₀=0 for P−P_(eq)≧ΔP_(threshold). $\begin{matrix}        {{N_{0} \propto {{\exp\left( {- \frac{\delta}{\Delta\quad P_{threshold}}} \right)} - {\exp\left( {- \frac{\delta}{\Delta\quad P_{threshold}}} \right)}}}{{{{for}\quad P} - P_{eq}} \leq {\Delta\quad P_{threshold}}}} & (20)        \end{matrix}$

Now, from Equation (16), h_(v) depends on N₀:h _(v) ≈aDN ₀ ^(2/3) S _(G) ^(2/3)  (21)

As mentioned above, exponent ⅔ results from the surface/volume ratio ofthe bubbles and it can be modified to take into account of a branched(fractal) shape of the bubbles in the porous medium. Therefore replacenext by a more general exponent d occurs if necessary. $\begin{matrix}{{h_{v}\left( S_{g} \right)} = {S_{g}^{d}\beta\quad{D\left\lbrack {{\exp\left( \frac{\delta}{P - P_{eq}} \right)} - {\exp\left( \frac{\delta}{\Delta\quad P_{threshod}} \right)}} \right\rbrack}^{d}}} & (22)\end{matrix}$

Since this model shows the size distribution of the nucleation sites,constants d and β have to be the same for the same fluid and the samesample.

As already mentioned above, the convective effect has to be taken intoaccount; a term depending on the Peclet number is therefore added toh_(v) as follows: $\begin{matrix}{{Pe} = \frac{V\quad 1}{D}} & (23)\end{matrix}$  h _(v) =A+Bpe ^(α)  (24)

This is a model with adjustable parameters. It is more predictive thanthe model obtained by the pore-scale approach or by reservoirsimulators. There is only one set of parameters for a singleexperimental device (rock and fluids). Besides, this transfercoefficient has a real physical meaning in the same way as a capillarypressure curve, and it can therefore characterize a rock-fluid system inthe case of a solution gas drive process. This transfer curveh_(v)(S_(g)) is experimentally determined.

Gas Phase Flow

Discontinuous Gas Phase

If the mechanism of mobilization of the nodules of a non-wetting fluidby a second wetting fluid as the basis is taken, there is a criticaluntrapping size which corresponds to a threshold saturation denoted byS_(g) ^(mob). The trapped gas fraction is taken equal to S_(g) ^(mob).It is assumed that the mean velocity of the clusters is proportional tothat of the continuous fluid. Besides, it is coherent to assume thatthis flow will depend on the viscosity ratio of the two fluids. Thisallows using, for the same rock, the same proportionality coefficientfor two oils of different viscosity. The formulation implanted in thesimulator with these assumptions isf _(g) =Fμ _(g)/μ₀(S_(g) −S _(g) ^(mob))u _(o) for S_(g)>S_(g) ^(mob)f _(g)=0 for S_(g)<S_(g) ^(mob)  (25)

-   -   with F proportionality coefficient, μ gas and oil viscosities.

Continuous Gas Phase

From a saturation threshold value, denoted S_(g)* here, a fraction ofthe gas is connected, Darcy can then apply. The relative permeabilityused can be the relative permeability of a displacement experiment takenfor a saturation of (S_(g)−S_(g)*). It is then obtained for the gasflow: $\begin{matrix}\begin{matrix}{f_{g} = 0} & {{{for}\quad S_{g}} \leq S_{g}^{mob}} \\{f_{g} = {{c^{ste}\left( {S_{g} - S_{g}^{mob}} \right)}u_{o}}} & {{{for}\quad S_{g}^{*}} \geq S_{g} \geq S_{g}^{mob}} \\{f_{g} = {{{c^{ste}\left( {S_{g}^{*} - S_{g}^{mob}} \right)}u_{o}} + {\frac{k\quad{k_{rg}\left( {S_{g} - S_{g}^{*}} \right)}}{\mu_{g}}\frac{\partial P}{\partial x}}}} & {{{for}\quad S_{g}} \geq S_{g}^{*}}\end{matrix} & (26)\end{matrix}$

Oil Phase Flow

The oil phase being continuous, the Darcy formalism is applied thereto.The relative oil permeability will be determined in a displacementexperiment.

System of Equations

With the various mass balances for the oil, the gas and the lightelements concentration in the oil, it is obtained:

For the oil: $\begin{matrix}{{{\Phi\frac{\partial}{\partial t}\left( {\rho_{0}S_{0}} \right)} + {\frac{\partial}{\partial x}\left( {\rho_{0}u_{0}} \right)}} = 0} & (26)\end{matrix}$

For the gas: $\begin{matrix}{{{\Phi\frac{\partial}{\partial t}\left( {PS}_{g} \right)} + {\frac{\partial}{\partial x}\left( {Pf}_{g} \right)}} = {\Phi\quad{{RTh}_{v}\left( S_{g} \right)}\left( {C - {k_{s}P}} \right)}} & (27)\end{matrix}$

For the concentration in the oil: $\begin{matrix}{{{\Phi\frac{\partial}{\partial t}\left( {CS}_{0} \right)} + {\frac{\partial}{\partial x}\left( {Cu}_{0} \right)}} = {{- \Phi}\quad{h_{v}\left( S_{g} \right)}\left( {C - {k_{s}P}} \right)}} & (28)\end{matrix}$

In Equation (27), the pressure appears through the expression of the gasdensity, the gas being considered to be a perfect gas.

Adjustment of the Model to the Experimental Results

FIG. 6 shows simulation examples for a C₁-C₃-C₁₀ light oil. A goodagreement is obtained for the various depletion rates. The model hasbeen calibrated on the extreme depletion rates. The same parameters havebeen used for all of the simulations.

In order to confirm the validity of the model for viscous oils, twoseries of simulations were carried, without convective effects.

FIGS. 7 and 8 show the first series of simulations. In both cases, therock is the same, but the oils are different. Calibration has beenperformed on the two extreme rates of FIG. 7. The same set of parametershas been used for all of the simulations, only S_(g) ^(mob) isdifferent.

FIGS. 9 and 10 show that there is a good correlation between two seriesof experiments carried out from two different samples.

1) A method for forming a model allowing simulation of production by anunderground reservoir, under the effect of depletion, comprising: a)measuring respective volumes of oil and gas produced by rock samplesfrom the reservoir subjected to depletion, as well as relativepermeabilities of rock samples to hydrocarbons b) determining, by a gasfraction flow model, a volume transfer coefficient by means of apressure-dependent empirical function that is calibrated with referenceto previous measurements, from which distribution N(P) of nuclei thatcan be activated at a pressure P is deduced, c) while considering thatthe distribution N(P) of nuclei in the reservoir rocks is the same as adistribution of microbubbles deduced from the measurements, determining,by means of the gas fraction flow model, the numerical transfercoefficient that corresponds thereto in the reservoir at selecteddepletion rates, which allows prediction of relative permeabilities inthe reservoir and reservoir production. 2) A method as claimed in claim1, wherein the gas fraction flow model is described by a parametercharacterizing a force required for untrapping the bubbles, a parametercharacterizing a change of a gas phase to a continuous form, theparameters being determined by calibration from the measurements, and byvalues of the relative permeability to a continuous gas fraction.